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Multi-Curve Convexity CMS Pricing with Normal Volatilities and Basis Spreads in QuantLib
Sebastian Schlenkrich London, July 12, 2016
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1. CMS Payoff and Convexity Adjustment
2. Annuity Mapping Function as Conditional Expectation
3. Normal Model for CMS Coupons
4. Extending QuantLib‘s CMS Pricing Framework
5. Summary and References
Agenda
2016-07-12 | Multi-Curve Convexity
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CMS Payoff and Convexity Adjustment
2016-07-12 | Multi-Curve Convexity | CMS Payoff and Convexity Adjustment
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A forward swap rate is given as float leg over annuity
𝑆 𝑡 = ∑ 𝐿 𝑡 ⋅ 𝜏 ⋅ 𝑃(𝑡, 𝑇 )∑ 𝜏 ⋅ 𝑃(𝑡, 𝑇 )
( )
We consider a call on a (say 10y) swap rate 𝑆 𝑇 fixed at 𝑇,
𝑆 𝑇 − 𝐾 and paid at 𝑇 ≥ 𝑇
Payoff is evaluated under the annuity meassure
𝑉 𝑡 = 𝐴𝑛 𝑡 ⋅ 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇) ⋅ 𝑆 𝑇 − 𝐾
› Annuity meassure because swap rate dynamics are in principle available from swaption skew
› However, additional term 𝑃(𝑇, 𝑇 )/𝐴𝑛(𝑇) requires special treatment (convexity)
CMS coupons refer to swap rates (like swaptions) but pay at a single pay date (unlike swaptions)
Tenor basis enters CMS pricing via swap rates (Libor forward curve) and additional discount terms (OIS discount curve)
2016-07-12 | Multi-Curve Convexity | CMS Payoff and Convexity Adjustment (1/2)
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Sometimes it makes sense to split up in Vanilla payoff and convexity adjustment
𝑉 𝑡 = 𝑃(𝑡, 𝑇 ) ⋅ 𝐸 𝑆 𝑇 − 𝐾 + 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇)
𝐴𝑛 𝑡𝑃(𝑡, 𝑇 ) − 1 ⋅ 𝑆 𝑇 − 𝐾
What are the challenges for calculating the convexity adjustment?
» We know the dynamics of 𝑆 𝑇 (under the annuity meassure)
» We do not know the dynamics of 𝑃(𝑇, 𝑇 )/𝐴𝑛(𝑇)
» But it is reasonable to assume a very strong relation between 𝑆 𝑇 and 𝑃(𝑇, 𝑇 )/𝐴𝑛(𝑇)
CMS payoff may be decomposed into a Vanilla part and a remaining convexity adjustment part
Vanilla option Convexity adjustment
For CMS pricing we need to express 𝑃(𝑇, 𝑇 )/𝐴𝑛(𝑇) in terms of the swap rate 𝑆 𝑇 taking into account tenor basis
2016-07-12 | Multi-Curve Convexity | CMS Payoff and Convexity Adjustment (2/2)
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Annuity Mapping Function as Conditional Expectation
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation
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Consider the iterated expectation
𝐸 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇) ⋅ 𝑆 𝑇 − 𝐾 | 𝑆 𝑇 = 𝑠 = 𝐸 𝐸 𝑃(𝑇, 𝑇 )
𝐴𝑛(𝑇) | 𝑆 𝑇 = 𝑠 ⋅ 𝑆 𝑇 − 𝐾
Define the annuity mapping function
𝛼 𝑠, 𝑇 = 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇) | 𝑆 𝑇 = 𝑠
By construction 𝛼 𝑠, 𝑇 is deterministic in 𝑠. We can write 𝑉 𝑡 = 𝐴𝑛 𝑡 ⋅ 𝐸 𝛼 𝑆 𝑇 , 𝑇 ⋅ 𝑆 𝑇 − 𝐾
= 𝐴𝑛 𝑡 ⋅ 𝛼 𝑆 𝑇 , 𝑇 ⋅ 𝑆 𝑇 − 𝐾 ⋅ 𝑑𝑃 𝑆 𝑇
Conceptually, CMS pricing consists of three steps
1. Determine terminal distrubution 𝑑𝑃 𝑆 𝑇 of swap rate (in annuity meassure)
2. Specify a model for the annuity mapping function 𝛼 𝑠, 𝑇
3. Integrate payoff and annuity mapping function analytically (if possible) or numerically
Quotient 𝑃(𝑇, 𝑇 )/𝐴𝑛(𝑇) is expressed in terms of the swap rate 𝑆(𝑇) by means of an annuity mapping function
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (1/9)
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In general tenor basis and multi-curve pricing affects CMS pricing by two means
𝑉(𝑡) = 𝐴𝑛 𝑡 ⋅ 𝛼 𝑆 𝑇 , 𝑇 ⋅ 𝑆 𝑇 − 𝐾 ⋅ 𝑑𝑃 𝑆 𝑇
1. Vanilla swaption pricing
› Required to determine terminal distribution
› Use tenor forward curve and Eonia
discount curve to calculate forward swap
rate
2. Construction of annuity mapping function
› Relate Eonia discount factors (and annuity)
to swap rates based on tenor forward curve
(and Eonia discount curve)
Multi-curve pricing for CMS coupons requires a basis model to specify the relation between discount factors and forward swap rate
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (2/9)
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How does an annuity mapping function look like in practice?
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-50% -30% -10% 10% 30% 50%
alph
a(S(
T),T
_p)
swap rate S(T)
T_p = 10y T_p = 20y
0.000.020.040.060.080.100.120.140.160.18
10 12 14 16 18 20al
pha(
S(T)
,T_p
)pay time T_p (years)
S(T)=-10% S(T) = 10%
𝛼 ⋅, 𝑇 𝛼 𝑆 𝑇 ,⋅
» For the Hull White model an annuity mapping function 𝛼 𝑆 𝑇 , 𝑇 can easily be calculated
» This gives an impression of its functional form
The annuity mapping function shows less curvature in 𝑆- and 𝑇-direction. Thus it appears reasonable to apply linear approximations
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (3/9)
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Assume an affine functional relation for the annuity mapping function 𝛼 𝑠, 𝑇 = 𝑎 𝑇 ⋅ 𝑠 + 𝑏(𝑇 )
for suitable time-dependent functions 𝑎 𝑇 and 𝑏(𝑇 ). By construction there is a fundamental no-arbitrage condition for TSR models
𝐸 𝛼(𝑆 𝑇 , 𝑇 ) = 𝐸 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇) | 𝑆 𝑇 = 𝑠 = 𝐸 𝑃(𝑇, 𝑇 )
𝐴𝑛(𝑇) = 𝑃(𝑡, 𝑇 )𝐴𝑛(𝑡)
From definition of the linear TSR model we get 𝐸 𝛼(𝑆 𝑇 , 𝑇 ) = 𝐸 𝑎 𝑇 ⋅ 𝑆(𝑇) + 𝑏(𝑇 ) = 𝑎 𝑇 ⋅ 𝑆(𝑡) + 𝑏(𝑇 )
Thus
𝑏 𝑇 = 𝑃(𝑡, 𝑇 )𝐴𝑛(𝑡) − 𝑎 𝑇 ⋅ 𝑆(𝑡)
This yields a linear TSR model representation only in terms of function 𝑎 𝑇 as(1)
𝜶 𝒔, 𝑻𝒑 = 𝒂 𝑻𝒑 ⋅ 𝒔 − 𝑺(𝒕) + 𝑷(𝒕, 𝑻𝒑)𝑨𝒏(𝒕)
One modelling approach for the annuity mapping function is a linear terminal swap rate (TSR) model
Slope function 𝑎 𝑇 corresponds to 𝐺′(𝑅 ) in Hagan‘s Convexity Conundrums paper
Linear TSR models only differ in their specification of the slope function 𝑎 𝑇 .
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (4/9)
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Remember that 𝐴𝑛 𝑇 = ∑ 𝜏 ⋅ 𝑃(𝑇, 𝑇 ). For all realisations 𝑠 of future swap rates 𝑆(𝑇) we have
𝜏 ⋅ 𝛼 𝑠, 𝑇 = 𝐸 𝜏 ⋅ 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇) | 𝑆 𝑇 = 𝑠 = 𝐸 𝐴𝑛(𝑇)
𝐴𝑛(𝑇) | 𝑆 𝑇 = 𝑠 = 1
Applying the linear TSR model yields
𝜏 ⋅ 𝛼 𝑠, 𝑇 = 𝜏 ⋅ 𝑎 𝑇 ⋅ 𝑠 − 𝑆 𝑡 + 𝜏 ⋅ 𝑃 𝑡, 𝑇𝐴𝑛 𝑡 = 1
Thus additivity condition for slope function 𝑎 ⋅ becomes
𝜏 ⋅ 𝑎 𝑇 = 0
A further model-independent condition is given as additivity condition
So far, no-arbitrage and additivity condition only depend on OIS discount factors. That is tenor basis does not affect them
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (5/9)
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Tenor basis is modelled as deterministic spread on continuous compounded forward rates for various tenors
Maturity
Con
t. C
omp.
Rat
es
6m Euribor tenor curve with forward rates 𝐿 (𝑡; 𝑇 , 𝑇 )
Eonia/OIS discount curve with discount factor 𝑃(𝑡, 𝑇)
𝑓 (𝑡, 𝑇)
𝑓(𝑡, 𝑇) Deterministic spread relation between forward rates
𝑓 𝑡, 𝑇 = 𝑓 𝑡, 𝑇 + 𝑏(𝑇)
Deterministic Relation betwen forward Libor rates and OIS discount factors
1 + 𝜏 ⋅ 𝐿 𝑡 = 𝐷 ⋅ 𝑃 𝑡, 𝑇𝑃 𝑡, 𝑇 with 𝐷 = 𝑒∫
Swap rates may be expressed in terms of discount factors (without Libor rates)
𝑆 𝑡 = ∑ 𝐿 𝑡 ⋅ 𝜏 ⋅ 𝑃(𝑡, 𝑇 )∑ 𝜏 ⋅ 𝑃(𝑡, 𝑇 ) = ∑ 𝜔 ⋅ 𝑃(𝑡, 𝑇 )
∑ 𝜏 ⋅ 𝑃(𝑡, 𝑇 ) with 𝜔 =𝐷 , 𝑖 = 0
𝐷 − 1, 𝑖 = 1, … , 𝑁 − 1−1, 𝑖 = 𝑁
We use the multiplicative terms 𝐷 to describe tenor basis
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (6/9)
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We have for all realisations 𝑠 of future swap rates 𝑆(𝑇)
𝜔 ⋅ 𝛼 𝑠, 𝑇 = 𝐸 ∑ 𝜔 ⋅ 𝑃(𝑇, 𝑇 )∑ 𝜏 ⋅ 𝑃(𝑇, 𝑇 ) | 𝑆 𝑇 = 𝑠 = 𝐸 𝑆 𝑇 | 𝑆 𝑇 = 𝑠 = 𝑠
Applying the linear TSR model yields
𝜔 ⋅ 𝛼 𝑠, 𝑇 = 𝜔 ⋅ 𝑎(𝑇 ) ⋅ 𝑠 − 𝑆 𝑡 + 𝜔 ⋅ 𝑃 𝑡, 𝑇𝐴𝑛 𝑡 = 𝑠
Above equations yield consistency condition specifying slope of 𝑎 ⋅
𝜔 ⋅ 𝑎 𝑇 = 1
Additional consistency condition links today‘s forward swap rate to discount factors
Tenor basis enters coefficients 𝜔 (via spread terms 𝐷 ). Thus tenor basis has a slight effect of the slope of annuity mapping function in 𝑇-direction
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (7/9)
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Neccessary (additivity and consistency) conditions for a linear TSR model are
𝜏 ⋅ 𝑎 𝑇 = 0 and 𝜔 ⋅ 𝑎 𝑇 = 1
If we set 𝑎 𝑇 = 𝑢 ⋅ 𝑇 − 𝑇 + 𝑣 then we may directly solve for 𝑢 and 𝑣
𝑢 = −∑ 𝜏̅
∑ 𝜏̅ 𝑇 − 𝑇 ⋅ ∑ 𝜔 − ∑ 𝜔 𝑇 − 𝑇 ⋅ ∑ 𝜏̅
𝑣 =∑ 𝜏̅ 𝑇 − 𝑇
∑ 𝜏̅ 𝑇 − 𝑇 ⋅ ∑ 𝜔 − ∑ 𝜔 𝑇 − 𝑇 ⋅ ∑ 𝜏̅
Additivity and consistency condition may be combined to fully specify an affine annuity mapping function
There are more sophisticated approaches available to specify the annuity mapping function. However, to be fully consistent, they might need to be adapted to the consistency condition with tenor basis
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (8/9)
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Comparing annuity mapping function in Hull White and affine TSR model shows reasonable approximation for relevant domain
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alph
a(S(
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swap rate S(T)
T_p = 10y T_p = 20y
T_p = 10y (affine) T_p = 20y (affine)
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pha(
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,T_p
)
pay time T_p (years)
S(T)=-10% S(T) = 10%
S(T)=-10% (affine) S(T)=10% (affine)
𝛼 ⋅, 𝑇 𝛼 𝑆 𝑇 ,⋅
2016-07-12 | Multi-Curve Convexity | Annuity Mapping Function as Conditional Expectation (9/9)
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Normal Model for CMS Coupons
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons
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𝑉 𝑡 = 𝑃(𝑡, 𝑇 ) ⋅ 𝐸 𝑆 𝑇 − 𝐾 + 𝐸 𝑃(𝑇, 𝑇 )𝐴𝑛(𝑇)
𝐴𝑛 𝑡𝑃(𝑡, 𝑇 ) − 1 ⋅ 𝑆 𝑇 − 𝐾
( )
Replacing ( , )( ) by 𝛼 𝑠, 𝑇 = 𝑎 𝑇 ⋅ 𝑆(𝑇) − 𝑆(𝑡) + ( , )
( ) (conditional expectation) yields
𝐶𝐴 𝑡 = 𝐸 𝑎 𝑇 ⋅ 𝑆(𝑇) − 𝑆 𝑡 + 𝑃 𝑡, 𝑇𝐴𝑛 𝑡
𝐴𝑛 𝑡𝑃 𝑡, 𝑇 − 1 ⋅ 𝑆 𝑇 − 𝐾
= 𝑎 𝑇 ⋅ 𝐴𝑛 𝑡𝑃 𝑡, 𝑇 ⋅ 𝐸 𝑆(𝑇) − 𝑆 𝑡 ⋅ 𝑆 𝑇 − 𝐾
Applying linear TSR model to CMS instruments
We do have a specification for slope function How to solve for the expectation?
Solving for the expectation requires a model for the swap rate. Due to current low/negative interest rates we will apply a normal model.
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons (1/5)
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𝑆(𝑇) − 𝑆 𝑡 ⋅ 𝑆 𝑇 − 𝐾 = − 𝑆 𝑡 − 𝐾 ⋅ 𝑆 𝑇 − 𝐾 + 1 ⋅ 𝑆 𝑇 − 𝐾
Vanilla option may be priced with Bachelier‘s formula and implied normal volatility 𝜎
Abbreviating 𝜈 = 𝜎 𝑇 − 𝑡 and ℎ = 𝑆 𝑡 − 𝐾 /𝜈 yields 𝐸 𝑆 𝑇 − 𝐾 = 𝜈 ℎ ⋅ 𝑁 ℎ + 𝑁′(ℎ)
Reusing the Vanilla model assumptions yields for the power option (after some algebra…)
𝐸 1 ⋅ 𝑆 𝑇 − 𝐾 = 𝜈 ℎ + 1 ⋅ 𝑁 ℎ + ℎ𝑁′(ℎ)
Convexity adjustment becomes
𝐸 𝑆(𝑇) − 𝑆 𝑡 ⋅ 𝑆 𝑇 − 𝐾 = 𝜈 ⋅ 𝑁(ℎ)
Convexity adjustment for CMS calls consists of Vanilla option and power option
Convexity adjustment Vanilla option Power option
Normal model yields compact formula for CMS convexity adjustment
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons (2/5)
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CMS caplet
𝐶𝐴(𝑡) = 𝑎 𝑇 ⋅ 𝐴𝑛 𝑡𝑃 𝑡, 𝑇 ⋅ 𝜈 ⋅ 𝑁(ℎ)
CMS floorlets
𝐶𝐴 𝑡 = −𝑎 𝑇 ⋅ 𝐴𝑛 𝑡𝑃 𝑡, 𝑇 ⋅ 𝜈 ⋅ 𝑁(−ℎ)
CMS swaplets
𝐶𝐴 𝑡 = 𝑎 𝑇 ⋅ 𝐴𝑛 𝑡𝑃 𝑡, 𝑇 ⋅ 𝜈
Analogously we find normal model convexity adjustments for CMS floorlets and CMS swaplets (1)
(1) Normal model CMS convexity adjustment formulas are also stated in a preprint Version of Hagan 2003
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons (3/5)
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Example CMS convexity adjustments for June‘16 market data based on Normal model
10y x 2y
CMS rate
10y x 10y
CMS rate
Single Curve: » Calculate swaprate and
conv. adjustment only by 6m Euribor curve
Multi Curve: » Calculate swaprate and
conv. adjustment by 6m Euribor forward and Eonia discount curve
Affine: » Affine TSR model (with
basis spreads) Standard: » (linearised) standard yield
curve model (see Hag’03) Mean Rev. 10%: » (linearised) yield curve
model based on mean reverting shifts (mean rev. 10%) (see Hag’03)
1.554% 1.542% 1.542% 1.542%
0.076%0.078% 0.078% 0.080%
1.50%1.52%1.54%1.56%1.58%1.60%1.62%1.64%
Single Curve -Affine
Multi Curve -Affine
Multi Curve -Standard
Multi Curve -Mean Rev. 10%
CMS
Rate
Index Fixing Conv. Adj.
1.493% 1.483% 1.483% 1.483%
0.256% 0.264% 0.276% 0.314%
1.40%1.45%1.50%1.55%1.60%1.65%1.70%1.75%1.80%1.85%
Single Curve -Affine
Multi Curve -Affine
Multi Curve -Standard
Multi Curve -Mean Rev. 10%
CMS
Rate
Index Fixing Conv. Adj.
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons (4/5)
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Model-implied 10y CMS swap spreads of Normal model show reasonable fit to quoted market data (1)
(1) Quotation 10y CMS swap spread: 10y CMS rate vs. 3m Euribor + quoted spread
40
60
80
100
120
140
5Y 10Y 15Y 20Y
ICAP Bid ICAP Ask
lognormal, standard yc model normal, standard yc model
normal, mean rev. 10% normal, affine yc model
2016-07-12 | Multi-Curve Convexity | Normal Model for CMS Coupons (5/5)
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Extending QuantLib‘s CMS Pricing Framework
2016-07-12 | Multi-Curve Convexity | Extending QuantLib‘s CMS Pricing Framework
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There is a flexible framework for CMS pricing in QuantLib which can easily be extended
FloatingRateCoupon
setPricer(…) FloatingRateCouponPricer
CmsCoupon CmsCouponPricer
HaganPricer LinearTsrPricer
Andersen/ Piterbarg 2010
Hagan 2003
We focus on the framework specified in the HaganPricer class
2016-07-12 | Multi-Curve Convexity | Extending QuantLib‘s CMS Pricing Framework (1/2)
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We add analytic formulas for Normal dynamics and affine TSR model with basis spreads
GFunction annuity mapping function class HaganPricer
CMS framework in QuantLib allows easy modifications and extensions, e.g., generalising NumericHaganPricer to normal or shifted log-normal volatilities
NumericHaganPricer static replication via Vanilla option pricer
AnalyticHaganPricer Black model based formulas
AnalyticNormalHaganPricer Bachelier model based formulas
GFunctionStandard bond-math based street standard model
GFunctionWithShift mean-reverting yield curve model
GFunctionAffine affine TSR model with basis spreads
2016-07-12 | Multi-Curve Convexity | Extending QuantLib‘s CMS Pricing Framework (2/2)
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Summary and References
2016-07-12 | Multi-Curve Convexity | Summary and References
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» Current low interest rates market environment requires generalisation of classical log-normal based CMS
convexity adjustment formulas
» Normal model for CMS pricing is easily be incorporated into QuantLib and yields good fit to CMS swap
quotes
» Tenor basis impacts specification of TSR models – however modelling effect is limited compared to other
factors
References
» P. Hagan. Convexity conundrums: pricing cms swaps, caps and floors. Wilmott Magazine, pages 3844,
March 2003.
» L. Andersen and V. Piterbarg. Interest rate modelling, volume I to III. Atlantic Financial Press, 2010.
» S. Schlenkrich. Multi-curve convexity. 2015. http://ssrn.com/abstract=2667405
» https://github.com/sschlenkrich/quantlib-old
Summary
2016-07-12 | Multi-Curve Convexity | Summary and References (1/1)
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d-fine GmbH
Frankfurt München London Wien Zürich
Zentrale
d-fine GmbH Opernplatz 2 D-60313 Frankfurt/Main
Tel +49 69 90737-0 Fax +49 69 90737-200
www.d-fine.com
Dr. Sebastian Schlenkrich Manager Tel +49 89-79086-170 Mobile +49 162-263-1525 E-Mail [email protected] Dr. Mark W. Beinker Partner Tel +49 69-90737-305 Mobile +49 151-14819305 E-Mail [email protected]
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